Student Journal - Ch7.pdf

Chapter

7

Fair Game Review

Complete the number sentence with < >, , or =.

1. 3.4 ______ 3.45 2. −6.01 ______ −6.1

3. 3.50 ______ 3.5 4. −0.84 ______ −0.91

Find three decimals that make the number sentence true. 5. −5.2 ≥ ______ 6. 2.65 > ______

7. −3.18 ≤ ______ 8. 0.03 < ______

9. The table shows the times of a 100-meter dash.

Chapter

7

Fair Game Review

Evaluate the expression.

10. ₁₀2 _{− 48 ÷ +6 25 • 3 11. 8} 16 ₂2 _{11 3}

4

  _{+ − •}  

 

12.

2

6

4 4 7

3

 ₊  _{÷ •}

 

  13.

(

)

2 ₂

5 9 − 4 − 3

14. ₅2 _{− 2}2 _{• 4}2 _{−12 15.} 2

2

50 4 5

  _÷    

16. The table shows the numbers of students in 4 classes. The teachers are combining the classes and dividing the students in half to form two groups for a project. Write an expression to represent this situation. How

7.1

Finding Square Roots

Essential Question

When you multiply a number by itself, you square the number.

2

4 4 4

16 = •

= 4 squared is 16.

To “undo” this, take the square root of the number.

2

16 ₌ 4 ₌ 4 The square root of 16 is 4.

Work with a partner. Use a square root symbol to write the side length of the square. Then find the square root. Check your answer by multiplying.

a. Sample: s ₌ 121 ₌ Check:

The length of each side of the square is ______________________.

b. c. d.

1 ACTIVITY: Finding Square Roots

Symbol for squaring is the exponent 2.

s

s

Area = 121 ft2

s

s

Area = 81 yd2

s

s

Area = 324 cm2

s

s

Area = 361 mi2

7.1 Finding Square Roots (continued)

e. f. g.

Work with a partner. Find the radius of each circle.

a. b.

c. d.

Work with a partner.

The period of a pendulum is the time (in seconds) it takes the pendulum to swing back and forth.

2 ACTIVITY: Using Square Roots

s

s

Area = 225 mi2

s

s

Area = 2.89 in.2

s

s

Area = ft4 2

9

Area= 36πin.2

r

Area= πyd2

r

Area= 0.25πft2

r r

Area ₌ m9 2

16p

7.1 Finding Square Roots(continued)

What Is Your Answer?

4. IN YOUR OWN WORDS How can you find the dimensions of a square or circle when you are given its area? Give an example of each. How can you check your answers?

L 1.00 1.96 3.24 4.00 4.84 6.25 7.29 7.84 9.00

T

1 2 3

0 4 5 6 7 8 9 L

T

2 3 4 5 6 7 8

1

0

Length (feet)

Period (seconds)

7.1

Practice

Find the two square roots of the number.

1. 16 2. 100 3. 196

Find the square root(s).

4. 169 5. 4

225 6. − 12.25

Evaluate the expression.

7. 2 36 + 9 8. 8 11 25

121

− 9. 3 125 8

5

 

−

 

 

 

7.2

Finding Cube Roots

Essential Question

When you multiply a number by itself twice, you cube the number.

3

4 4 4 4

64 = • •

= 4 cubed is 64.

To “undo” this, take the cube root of the number.

3

3 _{64 = 4}3 _{= 4 The cube root of 64 is 4.}

Work with a partner. Use a cube root symbol to write the edge length of the cube. Then find the cube root. Check your answer by multiplying.

a. Sample: _{s =} 3 _{343 =} 3 ₇3 _{= 7 inches}

The edge length of the cube is 7 inches.

b. c.

1 ACTIVITY: Finding Cube Roots

Symbol for cube root is 3 _. Symbol for cubing is the exponent 3.

Check

7 7 7 49 7 343 • • = • =  s s s

Volume = 343 in.3

s s

s

Volume ₌ 27 ft3

s s

s

7.2 Finding Cube Roots (continued)

d. e.

Work with a partner. Write the prime factorization of each number. Then use the prime factorization to find the cube root of the number.

a. 216

(

) (

) (

)

216 3 2 3 3 2 2

3 3 3 = • • • • •

= • • • • •

= • •

The cube root of 216 is __________.

b. 1000 c. 3375

2 ACTIVITY: Use Prime Factorizations to Find Cube Roots s

s

s

Volume = 0.001 cm3

s s

s Volume = yd1 3

8 2 216 3 72 8 9 4 2 2 • • • •

3 • 3

Prime factorization

Commutative Property of Multiplication

7.2 Finding Cube Roots(continued)

d. STRUCTURE Does this procedure work for every number? Explain why or why not.

What Is Your Answer?

3. Complete each statement using positive or negative.

a. A positive number times a positive number is a ____________ number.

b. A negative number times a negative number is a ___________ number.

c. A positive number multiplied by itself twice is a ____________ number.

d. A negative number multiplied by itself twice is a ___________ number.

4. REASONING Can a negative number have a cube root? Give an example to support your explanation.

5. IN YOUR OWN WORDS How is the cube root of a number different from the square root of a number?

6. Give an example of a number whose square root and cube root are equal.

7.2

Practice

Find the cube root.

1. 3 _{27 2.} 3 ₈

3. 3 _{−64 4.} 3 125

216 −

Evaluate the expression.

5. 10 −

( )

7.3

The Pythagorean Theorem

Essential Question

Pythagoras was a Greek mathematician and philosopher who discovered one of the most famous rules in

mathematics. In mathematics, a rule is called a theorem. So, the rule that Pythagoras discovered is called the Pythagorean Theorem.

Work with a partner.

a. On grid paper, draw any right triangle. Label the lengths of the two shorter sides a and b.

b. Label the length of the longest side c.

c. Draw squares along each of the three sides. Label the areas of the three squares a2, b2, and c2.

d. Cut out the three squares. Make eight copies of the right triangle and cut them out. Arrange the figures to form two identical larger squares.

e. MODELING The Pythagorean Theorem describes the relationship among a2, b2, and c2. Use your result from part (d) to write an equation that describes this relationship.

1 ACTIVITY: Discovering the Pythagorean Theorem

Pythagoras (c. 570–c. 490 B.C.)

7.3 The Pythagorean Theorem (continued)

Work with a partner. Use a ruler to measure the longest side of each right triangle. Verify the result of Activity 1 for each right triangle.

a. b.

c. d.

2 ACTIVITY: Using the Pythagorean Theorem in Two Dimensions

3 cm 4 cm

2 cm

4.8 cm

3 in. 1 in.1

4

2 in. 1 in.1

7.3 The Pythagorean Theorem(continued)

Work with a partner. A guy wire attached 24 feet above ground level on a telephone pole provides support for the pole.

a. PROBLEM SOLVING Describe a procedure that you could use to find the length of the guy wire without directly measuring the wire.

b. Find the length of the wire when it meets the ground 10 feet from the base of the pole.

What Is Your Answer?

4. IN YOUR OWN WORDS How are the lengths of the sides of a right triangle related? Give an example using whole numbers.

3 ACTIVITY: Using the Pythagorean Theorem in Three Dimensions

7.3

Practice

Find the missing length of the triangle.

1. 2. 3.

Find the missing length of the figure.

4. 5.

6. In wood shop, you make a bookend that is in the shape of a right triangle. What is the base b of the bookend?

c

28

21 a

17

15 _b 7.3

4.8

x

63 cm

16 cm

x

35 m

13 m

5 m

8 in.

7.4

Approximating Square Roots

Essential Question

Work with a partner. Archimedes was a Greek mathematician, physicist, engineer, inventor, and astronomer. He tried to find a rational number

whose square is 3. Two that he tried were 265and .1351

153 780

a. Are either of these numbers equal to 3? Explain.

b. Use a calculator to approximate 3. Write the number on a piece of paper. Enter it into the calculator and square it. Then subtract 3. Do you get 0? What does this mean?

c. The value of 3is between which two integers?

d. Tell whether the value of 3is between the given numbers. Explain your reasoning.

Work with a partner. Refer to the square on the number line below.

a. What is the length of the diagonal of the square?

b. Copy the square and its diagonal onto a piece of transparent paper. Rotate it about zero on the number line so that the diagonal aligns with the number

1 ACTIVITY: Approximating Square Roots

1.7 and 1.8 1.72 and 1.73 1.731 and 1.732

2 ACTIVITY: Approximating Square Roots Geometrically

7.4 Approximating Square Roots(continued)

c. STRUCTURE How do you think your answers in parts (a) and (b) are related?

Work with a partner.

a. Use grid paper and the given scale to draw a horizontal line segment 1 unit in length. Draw your segment near the bottom of the grid. Label this segment AC.

b. Draw a vertical line segment 2 units in length. Draw your segment near the left edge of the grid. Label this segment DC.

c. Set the point of a compass on A. d. Use the Pythagorean Theorem to Set the compass to 2 units. Swing find the length of segment BC. the compass to intersect segment

DC. Label this intersection as B.

7.4 Approximating Square Roots(continued)

4. Compare your approximation in Activity 3 with your results from Activity 1.

What Is Your Answer?

5. Repeat Activity 3 for a triangle in which segment AC is 2 units and segment BA is 3 units. Use the Pythagorean Theorem to find the length of segment BC. Use the grid paper to approximate 5 to the nearest tenth.

6. IN YOUR OWN WORDS How can you find decimal approximations of square roots that are not rational?

Scale:

7.4

Practice

Classify the real number.

1. 14 2. 3

7

− 3. 153

3

Estimate the square root to the nearest (a) integer and (b) tenth.

4. 8 5. 60 6. 172

25 −

Which number is greater? Explain.

7. 88, 12 8. ₋ 18, 6₋ 9. 14.5, 220

Practice

For use after Extension 7.4 Extension

7.4

Write the decimal as a fraction or a mixed number.

1. 0.3 2. −0.2

3. 1.7 4. −2.6

E x t e n si o n

7.4 Practice (continued)

7. −0.73 8. 0.18

9. −3.24 10. 1.09

7.5

Using the Pythagorean Theorem

Essential Question

The converse of a statement switches the hypothesis and the conclusion.

Work with a partner. Write the converse of the true statement. Determine whether the converse is true or false. If it is true, justify your reasoning. If it is false, give a counterexample.

a. If _{a = b, then a}2 _{= b}2_.

Converse:__________________________________________________

b. If _{a = b, then a}3 _{= b}3_.

Converse:__________________________________________________

c. If one figure is a translation of another figure, then the figures are congruent.

Converse:__________________________________________________

d. If two triangles are similar, then the triangles have the same angle measures.

Converse:__________________________________________________

Is the converse of a true statement always true? always false? Explain.

1 ACTIVITY: Analyzing Converses of Statements

Statement: If p, then q.

7.5 Using the Pythagorean Theorem (continued)

Work with a partner. The converse of the Pythagorean Theorem states: “If the equation _a2 _{+ b}2 _{= c}2 _{is true for the side lengths of a triangle, then the} triangle is a right triangle.”

a. Do you think the converse of the Pythagorean Theorem is true or false? How could you use deductive reasoning to support your answer?

b. Consider





• What does the Pythagorean Theorem tell you about



• What does this tell you about c and x?

• What does this tell you about



and



• What does this tell you about ∠E?

2 ACTIVITY: The Converse of the Pythagorean Theorem

c

b a

E F

D

x

b a

K L

7.5 Using the Pythagorean Theorem (continued)

Work with a partner. Follow the steps below to write a formula that you can use to find the distance between and two points in a coordinate plane.

Step 1: Choose two points in the coordinate plane that do not lie on the same horizontal or vertical line. Label the points

(

)

(

)

Step 2: Draw a line segment connecting the points. This will be the hypotenuse of a right triangle.

Step 3: Draw horizontal and vertical line segments from the points to form the legs of the right triangle.

Step 4: Use the x-coordinates to write an expression for the length of the horizontal leg.

Step 5: Use the y-coordinates to write Step 6: Substitute the expressions for an expression for the length of the lengths of the legs into the vertical leg. the Pythagorean Theorem.

Step 7: Solve the equation in Step 6 for the hypotenuse c.

What does the length of the hypotenuse tell you about the two points?

What Is Your Answer?

4. IN YOUR OWN WORDS In what other ways can you use the Pythagorean Theorem?

5. What kind of real-life problems do you think the converse of the Pythagorean Theorem can help you solve?

3 ACTIVITY: Developing the Distance Formula

−2 −3 −4 −5 1 2 3 4 5 x 5 4 3 2 1 −2 −3 −4

7.5

Practice

Tell whether the triangle with the given side lengths is a right triangle.

1. 2.

3. 4 m, 4.2 m, 5.8 m 4. 31 in., 35 in., 16 in.

Find the distance between the two points.

5.

( ) (

)

(

) ( )

7.

(

) (

)

(

) (

)

9. The cross-section of a wheelchair ramp is shown. Does the ramp form a right triangle?

18 yd

10 yd 14 yd

26 mm